Theorem ipsstr

Description: Lemma to shorten proofs of ipsbase through ipsvsca . (Contributed by Stefan O'Rear, 27-Nov-2014) (Revised by Mario Carneiro, 29-Aug-2015) (Revised by Thierry Arnoux, 16-Jun-2019)

	Ref	Expression
Hypothesis	ipspart.a	$$⊢A=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{\times }⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),S⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),I⟩\right\}$$
Assertion	ipsstr	$$⊢A\mathrm{Struct}⟨1,8⟩$$

Proof

Step	Hyp	Ref	Expression
1		ipspart.a	$$⊢A=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{\times }⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),S⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),I⟩\right\}$$
2		eqid	$$⊢\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{\times }⟩\right\}=\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{\times }⟩\right\}$$
3	2	rngstr	$$⊢\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{\times }⟩\right\}\mathrm{Struct}⟨1,3⟩$$
4		5nn	$$⊢5\in \mathbb{N}$$
5		scandx	$$⊢\mathrm{Scalar}\left(\mathrm{ndx}\right)=5$$
6		5lt6	$$⊢5<6$$
7		6nn	$$⊢6\in \mathbb{N}$$
8		vscandx	$$⊢{\cdot }_{\mathrm{ndx}}=6$$
9		6lt8	$$⊢6<8$$
10		8nn	$$⊢8\in \mathbb{N}$$
11		ipndx	$$⊢{\cdot }_{𝑖}\left(\mathrm{ndx}\right)=8$$
12	4 5 6 7 8 9 10 11	strle3	$$⊢\left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),S⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),I⟩\right\}\mathrm{Struct}⟨5,8⟩$$
13		3lt5	$$⊢3<5$$
14	3 12 13	strleun	$$⊢\left(\left\{⟨{\mathrm{Base}}_{\mathrm{ndx}},B⟩,⟨{+}_{\mathrm{ndx}},\underset{\dot{}}{+}⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{\times }⟩\right\}\cup \left\{⟨\mathrm{Scalar}\left(\mathrm{ndx}\right),S⟩,⟨{\cdot }_{\mathrm{ndx}},\underset{\dot{}}{·}⟩,⟨{\cdot }_{𝑖}\left(\mathrm{ndx}\right),I⟩\right\}\right)\mathrm{Struct}⟨1,8⟩$$
15	1 14	eqbrtri	$$⊢A\mathrm{Struct}⟨1,8⟩$$